# Quasitruncated cuboctahedron

Quasitruncated cuboctahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymQuitco
Coxeter diagramx4/3x3x ()
Elements
Faces12 squares, 8 hexagons, 6 octagrams
Edges24+24+24
Vertices48
Vertex figureScalene triangle, edge lengths 2, 3, 2–2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{13-6\sqrt2}}{2} ≈ 1.06239}$
Volume${\displaystyle 2(11-7\sqrt2) ≈ 2.20101}$
Dihedral angles8/3–4: 135°
8/3–6: ${\displaystyle \arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561^\circ}$
6–4: ${\displaystyle \arccos\left(\frac{\sqrt6}{3}\right) ≈ 35.26439^\circ}$
Central density-1
Number of external pieces146
Level of complexity26
Related polytopes
ArmySemi-uniform Girco, edge lengths ${\displaystyle \sqrt2-1}$ (octagons), ${\displaystyle 2-\sqrt2}$ (ditrigon-rectangle)
RegimentQuitco
DualGreat disdyakis dodecahedron
ConjugateGreat rhombicuboctahedron
Convex coreOctahedron
Abstract & topological properties
Flag count288
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryB3, order 48
ConvexNo
NatureTame

The quasitruncated cuboctahedron or quitco, also called the great truncated cuboctahedron, is a uniform polyhedron. It consists of 12 squares, 8 hexagons, and 6 octagrams, with one of each type of face meeting per vertex. It can be obtained by quasicantitruncation of the cube or octahedron, or equivalently by quasitruncating the vertices of a cuboctahedron and then adjusting the edge lengths to be all equal.

## Vertex coordinates

A quasitruncated cuboctahedron of edge length 1 has vertex coordinates given by all permutations of:

• ${\displaystyle \left(±\frac{2\sqrt2-1}{2},\,±\frac{\sqrt2-1}{2},\,±\frac12\right).}$

## Related polyhedra

o4/3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cube cube {4/3,3} x4/3o3o ()
Quasitruncated hexahedron quith t{4/3,3} x4/3x3o ()
Cuboctahedron co r{3,4/3} o4/3x3o ()
Truncated octahedron toe t{3,4/3} o4/3x3x ()
Octahedron oct {3,4/3} o4/3o3x ()
Quasirhombicuboctahedron querco rr{3,4/3} x4/3o3x ()
Quasitruncated cuboctahedron quitco tr{3,4/3} x4/3x3x ()
(degenerate, oct+6(4)) o4/3o3ß ()
Icosahedron ike s{3,4/3} o4/3s3s ()